Yes, you can unzip the TAR file and extract the files into your root folder where you want WordPress installed. But that is only half of your setup. The other half is the database.
You will need to export your database from the current site and then import it into your new site. A lot of hosts offer an online tool called phpMyAdmin which you can use to export and import the database.
After that is done you may need to change the site url, if that is changing.
There are plenty of guides online that will walk you through step by step if you need it.
There are also plenty of plugins out there that will help with the site migration, just search for “WordPress Migration Plugin”.